4.1 Mass

Consider again the configuration of body $\mathcal{B}$ at time t (i.e., the present configuration) in which $\mathcal{B}$ occupies the region $\mathcal{R}$ of ${\mathcal{E}}^3$ bounded by the closed surface $\partial \mathcal{R}$; a part $\mathcal{S}$ of body $\mathcal{B}$ occupies a region $\mathcal{P}\subseteq \mathcal{R}$ bounded by a closed surface $\partial \mathcal{P}$; refer to Figure 3.9. Each part $\mathcal{S}$ of body $\mathcal{B}$ at each instant of time is assumed to be endowed with a nonnegative measure $\mathcal{M}\left(\mathcal{S},t\right)$, called the mass of part $\mathcal{S}$.

We define the mass density ρ at particle Y in the present configuration by

$\rho =\overline{\rho }\left(Y,t\right)=\underset{V\to 0}{\text{lim}}\frac{\mathcal{M}\left(\mathcal{S},t\right)}{V},$

where $V=V\left(\mathcal{P}\right)$ is the volume of the region $\mathcal{P}$ occupied by subset $\mathcal{S}$ as it collapses to particle Y. Since V is positive and $\mathcal{M}\left(\mathcal{S},t\right)$ is nonnegative, density ρ is nonnegative.

In continuum mechanics, it is assumed that the measure $\mathcal{M}\left(\mathcal{S},t\right)$ is absolutely continuous, i.e., in every configuration a part $\mathcal{S}$ occupying a sufficiently small volume has arbitrarily small mass $\mathcal{M}\left(\mathcal{S},t\right)$. Thus, concentrated point, line, and surface masses are excluded, and the limit (4.1) always exists.

The mass of part $\mathcal{S}$ of body $\mathcal{B}$ and the mass of body $\mathcal{B}$ itself at time t can be expressed in terms of density ρ by

$\mathcal{M}\left(\mathcal{S},t\right)={\displaystyle \underset{\mathcal{P}}{\int }\rho \text{d}v},\mathcal{M}\left(\mathcal{B},t\right)={\displaystyle \underset{\mathcal{R}}{\int }\rho \text{d}v},$

where dv is an element of volume in the present configuration (refer to Section 3.6).

We define the mass density ρ_R at particle Y in the reference configuration by

${\rho }_{\text{R}}={\overline{\rho }}_{\text{R}}\left(Y\right)=\underset{V_{\text{R}}\to 0}{\text{lim}}\frac{\mathcal{M}\left({\mathcal{S}}_{\text{R}}\right)}{V_{\text{R}}},$

where $V_{\text{R}}=V_{\text{R}}\left({\mathcal{P}}_{\text{R}}\right)$ is the volume of region ${\mathcal{P}}_{\text{R}}$ occupied by subset $\mathcal{S}$ in the reference configuration, and $\mathcal{M}\left({\mathcal{S}}_{\text{R}}\right)$ is the mass of $\mathcal{S}$ in the reference configuration, as $\mathcal{S}$ collapses to particle Y.

The mass of part $\mathcal{S}$ of body $\mathcal{B}$ and the mass of body $\mathcal{B}$ itself in the reference configuration can be expressed in terms of the reference density ρ_R by

$\mathcal{M}\left({\mathcal{S}}_{\text{R}}\right)={\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }{\rho }_{\text{R}}\text{d}V},\mathcal{M}\left({\mathcal{B}}_{\text{R}}\right)={\displaystyle \underset{{\mathcal{R}}_{\text{R}}}{\int }{\rho }_{\text{R}}\text{d}V},$

where dV is an element of volume in the reference configuration (refer to Section 3.6).

The densities ρ and ρ_R, defined in (4.1) and (4.3), respectively, both have material, Lagrangian, and Eulerian descriptions:

$\begin{array}{l}\rho =\overline{\rho }\left(Y,t\right)=\overline{\rho }\left({\kappa }^{-1}\left(\mathbf{X}\right),t\right)=\widehat{\rho }\left(\mathbf{X},t\right)=\widehat{\rho }\left({\chi }^{-1}\left(\mathbf{x},t\right),t\right)=\tilde{\rho }\left(\mathbf{x},t\right),\hfill \\ {\rho }_{\text{R}}={\overline{\rho }}_{\text{R}}\left(Y\right)={\overline{\rho }}_{\text{R}}\left({\kappa }^{-1}\left(\mathbf{X}\right)\right)={\widehat{\rho }}_{\text{R}}\left(\mathbf{X}\right)={\widehat{\rho }}_{\text{R}}\left({\chi }^{-1}\left(\mathbf{x},t\right)\right)={\tilde{\rho }}_{\text{R}}\left(\mathbf{x},t\right).\hfill \end{array}$

Note that since the material and Lagrangian descriptions of ρ_R have no time dependence,

${\stackrel{.}{\rho }}_{\text{R}}=\frac{\partial }{\partial t}{\overline{\rho }}_{\text{R}}\left(Y\right)=\frac{\partial }{\partial t}{\widehat{\rho }}_{\text{R}}\left(\mathbf{X}\right)=0.$

4.2 Forces and moments, linear and angular momentum

Again, consider the part $\mathcal{S}$ of body $\mathcal{B}$ that occupies region $\mathcal{P}$ bounded by surface $\partial \mathcal{P}$ in the present configuration at time t. We denote the resultant external force acting on part $\mathcal{S}$ of the body by f:

$\mathbf{f}=\mathbf{f}\left(\mathcal{S},t\right),\text{the resultant external force acting on part}\mathcal{S}\text{at time}t,$

and the resultant external moment about the origin (point 0) by M_o:

${\mathbf{M}}_{\mathbf{o}}={\mathbf{M}}_{\mathbf{o}}\left(\mathcal{S},t,\mathbf{0}\right),\text{the resultant external moment acting on part}\mathcal{S}\text{at time}t\text{about the origin}\mathbf{0}.$

We assume that the resultant external force f acting on part $\mathcal{S}$ at time t is composed of a body force f_b and a contact force f_c:

$\mathbf{f}={\mathbf{f}}_{\text{b}}+{\mathbf{f}}_{\text{c}}.$

Additionally, we make the following smoothness assumptions:

${\mathbf{f}}_{\text{b}}={\mathbf{f}}_{\text{b}}\left(\mathcal{S},t\right)={\displaystyle \underset{\mathcal{P}}{\int }\mathbf{b}\rho \text{d}v},{\mathbf{f}}_{\text{c}}={\mathbf{f}}_{\text{c}}\left(\mathcal{S},t\right)={\displaystyle \underset{\partial \mathcal{P}}{\int }\mathbf{t}\text{d}a},$

where

$\mathbf{b}=\tilde{\mathbf{b}}\left(\mathbf{x},t\right),\text{the}\mathbf{specific}\mathbf{body}\mathbf{force},\text{i}.\text{e}.,\text{the body force per unit mass}$

and

$\mathbf{t}=\tilde{\mathbf{t}}\left(\mathbf{x},t,\text{geometry of surface}\right),\text{the}\mathbf{traction},\text{i}.\text{e}.,\text{the contact force per unit area of the present configuration}.$

The geometry of the surface includes orientation, curvature, etc. The smoothness assumptions demand that b and t are bounded, continuous functions of space and time. We have already invoked a smoothness assumption on mass in the previous section, namely,

$\mathcal{M}\left(\mathcal{S},t\right)={\displaystyle \underset{\mathcal{P}}{\int }\rho \text{d}v},$

where density ρ is a bounded, continuous function of space and time. With this smoothness assumption on mass, we have excluded point, line, and surface masses; our smoothness assumptions on forces exclude concentrated forces.

We assume there are no distributed moment fields and no concentrated moments (we later relax this assumption in Chapter 9 to accommodate the effects of electro-magnetism), so the resultant external moment M_o on part $\mathcal{S}$ at time t about the origin 0 comes from only t and b:

${\mathbf{M}}_{\mathbf{o}}\left(\mathcal{S},t,\mathbf{0}\right)={\displaystyle \underset{\mathcal{P}}{\int }\left(\mathbf{x}-\mathbf{0}\right)\times \mathbf{b}\rho \text{d}v}+{\displaystyle \underset{\partial \mathcal{P}}{\int }\left(\mathbf{x}-\mathbf{0}\right)\times \mathbf{t}\text{d}a}.$

Because of our smoothness assumption on mass, the linear momentum $\mathcal{L}$ of part $\mathcal{S}$ at time t and the angular momentum H_o of part $\mathcal{S}$ about the origin at time t are also smooth:

$\mathbf{ℒ}=\mathbf{ℒ}\left(\mathcal{S},t\right)={\displaystyle \underset{\mathcal{P}}{\int }\mathbf{v}\rho \text{d}v},{\mathbf{H}}_{\mathbf{o}}={\mathbf{H}}_{\mathbf{o}}\left(\mathcal{S},t,\mathbf{0}\right)={\displaystyle \underset{\mathcal{P}}{\int }\left(\mathbf{x}-\mathbf{0}\right)\times \mathbf{v}\rho \text{d}v}.$

4.3 Equations of motion (mechanical conservation laws)

We postulate the following equations of motion:

• The mass of every subset $\mathcal{S}$ of the body remains constant throughout the motion, or, equivalently, the rate of change of the mass of $\mathcal{S}$ is zero (conservation of mass).

• The rate of change of linear momentum of $\mathcal{S}$ is equal to the resultant force acting on $\mathcal{S}$ (balance of linear momentum).

• The rate of change of angular momentum of $\mathcal{S}$ about the origin is equal to the resultant moment acting on $\mathcal{S}$ about the origin (balance of angular momentum).

Mathematically, these equations of motion can be expressed in material form as

$\frac{\text{d}}{\text{d}t}\mathcal{M}\left(\mathcal{S},t\right)=0\text{or}\mathcal{M}\left(\mathcal{S}\right)=\text{independent of}t=\mathcal{M}\left({\mathcal{S}}_{\text{R}}\right),$

$\frac{\text{d}}{\text{d}t}\mathbf{ℒ}\left(\mathcal{S},t\right)=\mathbf{f}\left(\mathcal{S},t\right),$

$\frac{\text{d}}{\text{d}t}{\mathbf{H}}_{\mathbf{o}}\left(\mathcal{S},t,\mathbf{0}\right)={\mathbf{M}}_{\mathbf{o}}\left(\mathcal{S},t,\mathbf{0}\right)$

for arbitrary part $\mathcal{S}$ of body $\mathcal{B}$, and all time t. We emphasize that the equations of motion (4.10a)–(4.10c) are global; i.e., they are valid not only for the body as a whole, but also for every arbitrary subset of the body. (The reader is already familiar with this requirement: as an example, in a static truss, not only is the entire structure in equilibrium, but so is each joint and each member.)

The smoothness assumptions discussed in Sections 4.1 and 4.2 can then be used to express the equations of motion (4.10a)₁, (4.10b), and (4.10c) in Eulerian integral form:

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{\mathcal{P}}{\int }\rho \text{d}v}=0,$

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{\mathcal{P}}{\int }\mathbf{v}\rho \text{d}v}={\displaystyle \underset{\mathcal{P}}{\int }\mathbf{b}\rho \text{d}v}+{\displaystyle \underset{\partial \mathcal{P}}{\int }\mathbf{t}\text{d}a},$

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{\mathcal{P}}{\int }\mathbf{x}\times \mathbf{v}\rho \text{d}v}={\displaystyle \underset{\mathcal{P}}{\int }\mathbf{x}\times \mathbf{b}\rho \text{d}v}+{\displaystyle \underset{\partial \mathcal{P}}{\int }\mathbf{x}\times \mathbf{t}\text{d}a}$

for arbitrary material volume $\mathcal{P}$ in the present configuration $\mathcal{R}$ of body $\mathcal{B}$, and all time t. To perform the integrations in (4.11a)–(4.11c) over areas and volumes in the present configuration, the functions ρ, v, b, and t must be in their Eulerian forms, i.e., functions of the independent variables x and t.

4.4 The first law of thermodynamics (conservation of energy)

To complete the set of fundamental laws of thermomechanics, we now postulate the law of conservation of energy, also known as the first law of thermodynamics. In general:

In this chapter, we specialize to thermomechanical systems, so the only other energy entering or leaving part $\mathcal{S}$ is heat. (This restriction is later relaxed in Chapter 9 to accommodate electromagnetic sources of energy.) It follows, then, that the law of conservation of energy in material form is

$\frac{\text{d}}{\text{d}t}T\left(\mathcal{S},t\right)=R\left(\mathcal{S},t\right)+H\left(\mathcal{S},t\right)$

for arbitrary subset $\mathcal{S}$ of body $\mathcal{B}$ and all time t, where

$T=T\left(\mathcal{S},t\right)$, the total energy of part $\mathcal{S}$ at time t,

$R=R\left(\mathcal{S},t\right)$, the rate of work done on part $\mathcal{S}$ at time t by the resultant external force f,

$H=H\left(\mathcal{S},t\right)$, the rate of heat energy entering part $\mathcal{S}$ at time t.

We emphasize that (4.12) is valid not only for the body as a whole, but also for every subset of the body.

The assumption (4.6) that the resultant external force f on $\mathcal{S}$ is the sum of a body force f_b and a contact force f_c implies that

$R=R_{\text{b}}+R_{\text{c}},$

and the smoothness assumptions on f_b and f_c (refer to (4.7)) further imply that

$R_{\text{b}}\left(\mathcal{S},t\right)={\displaystyle \underset{\mathcal{P}}{\int }\mathbf{b}·\mathbf{v}\rho \text{d}v}\left(\text{the rate of work of body forces}\right),$

$R_{\text{c}}\left(\mathcal{S},t\right)={\displaystyle \underset{\partial \mathcal{P}}{\int }\mathbf{t}·\mathbf{v}\text{d}a}\left(\text{the rate of work of contact forces}\right),$

where $\mathcal{P}$ is the region of ${\mathcal{E}}^3$ occupied by part $\mathcal{S}$ in the present configuration at time t, with boundary $\partial \mathcal{P}$.

Our smoothness assumption (4.2)₁ on mass implies that the energy of motion, or kinetic energy, of part $\mathcal{S}$ at time t is given by

$K\left(\mathcal{S},t\right)={\displaystyle \underset{\mathcal{P}}{\int }\frac{1}{2}\mathbf{v}·\mathbf{v}\rho \text{d}v}.$

We assume the existence of an internal energy E of part $\mathcal{S}$ of the body, such that

$T\left(\text{total energy}\right)=K+E\left(\text{kinetic energy}+\text{internal energy}\right),$

and further make the smoothness assumption

$E\left(\mathcal{S},t\right)={\displaystyle \underset{\mathcal{P}}{\int }\epsilon \rho \text{d}v},$

where

$\epsilon =\tilde{\epsilon }\left(\mathbf{x},t\right),\text{the}\mathbf{specific\; internal\; energy},\text{i.e., the internal energy per unit mass},$

is a bounded, continuous function (so we have excluded point, line, and surface concentrations of internal energy).

We assume that the rate of heat energy H entering part $\mathcal{S}$ at time t is composed of two parts, that entering throughout the volume $\mathcal{P}$ and that flowing through the surface $\partial \mathcal{P}$:

$H\left(\mathcal{S},t\right)={\displaystyle \underset{\mathcal{P}}{\int }r\rho \text{d}v}-{\displaystyle \underset{\partial \mathcal{P}}{\int }h\text{d}a},$

where

$r=\tilde{r}\left(\mathbf{x},t\right),\text{the}\mathbf{specific}\mathbf{heat}\mathbf{supply}\mathbf{rate},\text{i}.\text{e}.,\text{the heat energy absorbed per unit mass per unit time,}$

and

$h=\tilde{h}\left(\mathbf{x},t,\text{geometry of suface}\right),\text{the}\mathbf{heat}\mathbf{flux}\mathbf{rate},\text{i}.\text{e}.,\text{the heat flow}\mathit{out}\mathit{of}\partial \mathcal{P}\text{per unit area of the present configuration per unit time}.$

Physically, (4.18) represents heat transfer due to radiation (first term) and conduction (second term). Again, we make the smoothness assumption that r and h are bounded, continuous functions. The minus sign appears in (4.18) because of the sign convention that h is positive when heat flows out of $\partial \mathcal{P}$, i.e., through the surface $\partial \mathcal{P}$ in the direction of its outward unit normal n.

Using (4.13)–(4.18), we can express the first law of thermodynamics (4.12) in Eulerian integral form:

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{\mathcal{P}}{\int }\frac{1}{2}\mathbf{v}·\mathbf{v}\rho \text{d}v}+\frac{\text{d}}{\text{d}t}{\displaystyle \underset{\mathcal{P}}{\int }\epsilon \rho \text{d}v}={\displaystyle \underset{\mathcal{P}}{\int }\mathbf{b}·\mathbf{v}\rho \text{d}v}+{\displaystyle \underset{\partial \mathcal{P}}{\int }\mathbf{t}·\mathbf{v}\text{d}a}+{\displaystyle \underset{\mathcal{P}}{\int }r\rho \text{d}v}-{\displaystyle \underset{\partial \mathcal{P}}{\int }h\text{d}a}$

for arbitrary material volume $\mathcal{P}$ in the present configuration $\mathcal{R}$ of the body for all time t.

4.5 The transport and localization theorems

In this section, we present the transport theorem and the localization theorem. As will soon be evident, these are essential tools for rigorously deriving local (or pointwise) statements of the integral conservation laws developed in the previous two sections.

Exercises

1. Prove Leibniz's rule

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{a\left(t\right)}{\overset{b\left(t\right)}{\int }}\tilde{f}\left(x,t\right)\text{d}x}={\displaystyle \underset{a\left(t\right)}{\overset{b\left(t\right)}{\int }}\frac{\partial \tilde{f}\left(x,t\right)}{\partial t}\text{d}x}+\tilde{f}\left(b\left(t\right),t\right)\frac{\text{d}b\left(t\right)}{\text{d}t}-\tilde{f}\left(a\left(t\right),t\right)\frac{\text{d}a\left(t\right)}{\text{d}t}$

as the one-dimensional analog of the three-dimensional transport theorem. (Hint: First, define a one-dimensional motion x = χ(X, t), and define the limits A and B such that χ(A, t) = a(t) and χ(B, t) = b(t). Proceed as in the proof of the three-dimensional transport theorem shown in Problem 4.1.)

2. Using the definition of the material derivative and the divergence theorem, show that

${\displaystyle \underset{\mathcal{P}}{\int }\left(\stackrel{.}{\mathit{ɸ}}+\mathit{ɸ}\text{div}\mathbf{v}\right)\text{d}v}={\displaystyle \underset{\mathcal{P}}{\int }{\mathit{ɸ}}^{\prime }\text{d}v}+{\displaystyle \underset{\partial \mathcal{P}}{\int }\mathit{ɸ}\mathbf{v}·\mathbf{n}\text{d}a}.$

Then show that this result implies that

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{\mathcal{P}}{\int }\widetilde{\mathit{ɸ}}\left(\mathbf{x},t\right)\text{d}v}={\displaystyle \underset{\mathcal{P}}{\int }{\mathit{ɸ}}^{\prime }\text{d}v}+{\displaystyle \underset{\partial \mathcal{P}}{\int }\mathit{ɸ}\mathbf{v}·\mathbf{n}\text{d}a},$

which is an alternative form of the transport theorem.

4.6 Cauchy stress tensor, heat flux vector

Recall from Section 4.2 that the traction t acting on the surface $\partial \mathcal{P}$ is, in general, a function of position x, time t, and the geometry of the surface, i.e.,

$\mathbf{t}=\tilde{\mathbf{t}}\left(\mathbf{x},t,\text{geometry of surface}\right).$

The geometry of the surface includes orientation, curvature, and so on. We now restrict our attention to a particular type of contact force, namely, one such that the traction t is the same for all like-oriented surfaces with a common tangent plane at x and t. Therefore, the dependence of t on the geometry of the surface $\partial \mathcal{P}$ is only through the outward unit normal n, so (4.24) becomes

$\mathbf{t}=\tilde{\mathbf{t}}\left(\mathbf{x},t,\mathbf{n}\right).$

(An example of a traction that is not of the form (4.25) is surface tension, which depends on a measure of the curvature of the surface.)

We further assume that t is a continuous function of x, t, and n. Then, as a consequence of the conservation laws of mass (4.11a) and linear momentum (4.11b), and our assumptions that $\ddot{\mathbf{x}}$, b, and ρ are bounded, it can be shown (refer to Problem 4.3) that the dependence of t on n in (4.25) is in fact linear, i.e.,

$\tilde{\mathbf{t}}\left(\mathbf{x},t,\mathbf{n}\right)=\widehat{\mathbf{T}}\left(\mathbf{x},t\right)\mathbf{n}\text{or}\mathbf{t}=\mathbf{Tn}.$

The tensor T in (4.26) is called the Cauchy stress tensor. Note that the components of the Cauchy stress T are defined by

$T_{\mathit{ij}}=\mathbf{t}\left({\mathbf{e}}_j\right)·{\mathbf{e}}_i,$

where t(e_j) is the traction on a surface whose unit normal is n = e_j. This definition, along with result (4.26), implies that

$T_{\mathit{ij}}={\mathbf{e}}_i·\mathbf{T}{\mathbf{e}}_j,$

which is consistent with definition (2.29) of the Cartesian components of a tensor.

Problem 4.3

Prove that t = Tn.

Solution

Consider an arbitrary part $\mathcal{S}$ of body $\mathcal{B}$ that occupies a region $\mathcal{P}$ in the present configuration at time t. Let $\mathcal{P}$ be divided into two regions, ${\mathcal{P}}_1$ and ${\mathcal{P}}_2$, separated by a surface σ (see Figure 4.1). Define $\partial {\mathcal{P}}^{\prime }$ and $\partial {\mathcal{P}}^{\prime \prime }$ so that $\partial \mathcal{P}=\partial {\mathcal{P}}^{\prime }\cup \partial {\mathcal{P}}^{\prime \prime }$. Then, $\mathcal{P}={\mathcal{P}}_1\cup {\mathcal{P}}_2$, $\partial \mathcal{P}=\partial {\mathcal{P}}^{\prime }\cup \partial {\mathcal{P}}^{\prime \prime }$, $\partial {\mathcal{P}}_1=\partial {\mathcal{P}}^{\prime }\cup \sigma$, $\partial {\mathcal{P}}_2=\partial {\mathcal{P}}^{\prime \prime }\cup \sigma$.

Applying balance of linear momentum to the three regions $\mathcal{P}$, ${\mathcal{P}}_1$, and ${\mathcal{P}}_2$, we obtain

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{\mathcal{P}}{\int }\stackrel{.}{\mathbf{x}}\rho \text{d}v}={\displaystyle \underset{\mathcal{P}}{\int }\mathbf{b}\rho \text{d}v}+{\displaystyle \underset{\partial \mathcal{P}}{\int }\mathbf{t}\left(\mathbf{n}\right)\text{d}a},$

  (a)

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{{\mathcal{P}}_1}{\int }\stackrel{.}{\mathbf{x}}\rho \text{d}v}={\displaystyle \underset{{\mathcal{P}}_1}{\int }\mathbf{b}\rho \text{d}v}+{\displaystyle \underset{\partial {\mathcal{P}}^{\prime }\cup \sigma }{\int }\mathbf{t}\left(\mathbf{n}\right)\text{d}a},$

  (b)

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{{\mathcal{P}}_2}{\int }\stackrel{.}{\mathbf{x}}\rho \text{d}v}={\displaystyle \underset{{\mathcal{P}}_2}{\int }\mathbf{b}\rho \text{d}v}+{\displaystyle \underset{\partial {\mathcal{P}}^{″}\cup \sigma }{\int }\mathbf{t}\left(\mathbf{n}\right)\text{d}a}.$

  (c)

Adding (b) and (c) then subtracting (a) gives

${\displaystyle \underset{\sigma }{\int }\left[\mathbf{t}\left(\mathbf{n}\right)+\mathbf{t}\left(-\mathbf{n}\right)\right]\text{d}a}=\mathbf{0}.$

  (d)

In (d) we have noted that the outward normal of σ, when considered as a part of the boundary of ${\mathcal{P}}_1$, is directly opposed to the outward normal of σ when considered as a part of the boundary of ${\mathcal{P}}_2$. Assuming the traction is a continuous function of x and n, so that we can use the two-dimensional localization theorem, (d) implies

$\mathbf{t}\left(\mathbf{n}\right)=-\mathbf{t}\left(-\mathbf{n}\right).$

  (e)

(Traction vectors on opposite sides of the same surface are equal in magnitude, and opposite in direction.)

Consider now a family of similar tetrahedra $\mathcal{T}$ with heights h_p and a common vertex at some point _ox (see Figure 4.2). Sides i are perpendicular to the x_i directions and have outward normals −e_i. The remaining side has outward normal _on. From geometry, areas $\mathcal{S}$, ${\mathcal{S}}_1$, ${\mathcal{S}}_2$, and ${\mathcal{S}}_3$ are related by

${\mathcal{S}}_i=\mathcal{S}\left({}_{\mathbf{o}}\mathbf{n}·{\mathbf{e}}_i\right)=\mathcal{S}{n}_i.$

  (f)

The volumes of the tetrahedra are

$V=\frac{1}{3}{h}_{\text{p}}\mathcal{S}.$

  (g)

It is assumed that each of the tetrahedra lies completely within the region $\mathcal{R}$ occupied by $\mathcal{B}$ in the present configuration. By virtue of the transport theorem (4.20) and the local form of conservation of mass (presented later in Section 4.8), (a) becomes

${\displaystyle \underset{\mathcal{P}}{\int }\ddot{\mathbf{x}}\rho \text{d}v}={\displaystyle \underset{\mathcal{P}}{\int }\mathbf{b}\rho \text{d}v}+{\displaystyle \underset{\partial \mathcal{P}}{\int }\mathbf{t}\left(\mathbf{n}\right)\text{d}a}.$

  (h)

Applying (h) to a tetrahedron $\mathcal{T}$ gives

${\displaystyle \underset{\mathcal{T}}{\int }\left(\mathbf{x}\ddot{-}\mathbf{b}\right)\rho \text{d}v}={\displaystyle \underset{\text{Σ}{\mathcal{S}}_i}{\int }\mathbf{t}\left(-{\mathbf{e}}_i\right)\text{d}a}+{\displaystyle \underset{\mathcal{S}}{\int }\mathbf{t}\left({}_{\mathbf{o}}\mathbf{n}\right)\text{d}a}.$

  (i)

With use of (e), (i) becomes

${\displaystyle \underset{\mathcal{T}}{\int }\left(\mathbf{x}\ddot{-}\mathbf{b}\right)\rho \text{d}v}={\displaystyle \underset{\mathcal{S}}{\int }\mathbf{t}\left({}_{\mathbf{o}}\mathbf{n}\right)\text{d}a}-{\displaystyle \underset{\text{Σ}{\mathcal{S}}_i}{\int }\mathbf{t}\left({\mathbf{e}}_i\right)\text{d}a}.$

  (j)

Recall that the specific body force b and density ρ are assumed bounded throughout $\mathcal{R}$. The acceleration $\ddot{\mathbf{x}}$ is also assumed bounded. Then,

$\begin{array}{lll}\left|{\displaystyle \underset{\mathcal{T}}{\int }\left(\ddot{\mathbf{x}}-\mathbf{b}\right)\rho \text{d}v}\right|\hfill & \le {\displaystyle \underset{\mathcal{T}}{\int }\left|\ddot{\mathbf{x}}-\mathbf{b}\right|\rho \text{d}v}\hfill & \left(\text{by a theorem of analysis}\right)\hfill \\ \hfill & ={\displaystyle \underset{\mathcal{T}}{\int }K\text{d}v}\hfill & \left(\text{by boundedness}\right)\hfill \\ \hfill & =K{\displaystyle \underset{\mathcal{T}}{\int }\text{d}v}\hfill & \hfill \\ \hfill & =K\frac{\mathcal{S}{h}_{\text{p}}}{3}\hfill & \left(\text{using}\left(\text{g}\right)\right),\hfill \end{array}$

  (k)

where K is some number. Assuming that t is a continuous function of x and t, the mean value theorem and (f) imply that

${\displaystyle \underset{\mathcal{S}}{\int }\mathbf{t}\left({}_{\mathbf{o}}\mathbf{n}\right)\text{d}a}-{\displaystyle \underset{\text{Σ}{\mathcal{S}}_i}{\int }\mathbf{t}\left({\mathbf{e}}_i\right)\text{d}a}={\mathbf{t}}^{*}\left({}_{\mathbf{o}}\mathbf{n}\right)\mathcal{S}-{\mathbf{t}}^{*}\left({\mathbf{e}}_i\right){\mathcal{S}}_i=\left[{\mathbf{t}}^{*}\left({}_{\mathbf{o}}\mathbf{n}\right)-{\mathbf{t}}^{*}\left({\mathbf{e}}_i\right)n_i\right]\mathcal{S},$

  (l)

where t*(_on) and t*(e_i) stand for some specific interior values of the traction vectors on the faces $\mathcal{S}$ and ${\mathcal{S}}_i$, respectively. From (j) to (l),

$\begin{array}{ll}\frac{1}{3}K\mathcal{S}{h}_{\text{p}}\hfill & \ge \left|{\displaystyle \underset{\mathcal{T}}{\int }\left|\ddot{\mathbf{x}}-\mathbf{b}\right|\rho \text{d}v}\right|\hfill \\ \hfill & =\left|{\displaystyle \underset{\mathcal{S}}{\int }\mathbf{t}\left({}_{\mathbf{o}}\mathbf{n}\right)\text{d}a}-{\displaystyle \underset{\text{Σ}{\mathcal{S}}_i}{\int }\mathbf{t}\left({\mathbf{e}}_i\right)\text{d}a}\right|\hfill \\ \hfill & =\left|{\mathbf{t}}^{*}\left({}_{\mathbf{o}}\mathbf{n}\right)-{\mathbf{t}}^{*}\left({\mathbf{e}}_i\right)n_i\right|\mathcal{S},\hfill \end{array}$

so

$\left|{\mathbf{t}}^{*}\left({}_{\mathbf{o}}\mathbf{n}\right)-{\mathbf{t}}^{*}\left({\mathbf{e}}_i\right)n_i\right|\le \frac{1}{3}K{h}_{\text{p}}.$

As we consider smaller and smaller tetrahedra, h_p → 0, and

$\underset{h_{\text{p}}\to 0}{\text{lim}}\left|{\mathbf{t}}^{*}\left({}_{\mathbf{o}}\mathbf{n}\right)-{\mathbf{t}}^{*}\left({\mathbf{e}}_i\right)n_i\right|=0,$

  (m)

where t*(_on) and t*(e_i) are evaluated at the point _ox, which is the common vertex of the family of tetrahedra. From (m), we see that in the limit h_p → 0, t*(_on) − t*(e_i)n_i must be the zero vector, i.e.,

${\mathbf{t}}^{*}\left({}_{\mathbf{o}}\mathbf{n}\right)={\mathbf{t}}^{*}{\left({\mathbf{e}}_i\right)}_{\mathbf{o}}{n}_i,$

  (n)

where t*(e_i)_o denotes the value of t*(e_i) at the point _ox. Since (n) must hold at any point _ox and corresponding to any direction _on, without ambiguity, we may delete the star, replace _ox and _on by x and n, and write

$\mathbf{t}\left(\mathbf{n}\right)=\mathbf{t}\left({\mathbf{e}}_i\right)n_i.$

  (o)

We define T_ki by

$T_{\mathit{ki}}=\mathbf{t}\left({\mathbf{e}}_i\right)·{\mathbf{e}}_k,$

and denote the components of t(n) by

$t_k=\mathbf{t}·{\mathbf{e}}_k.$

Taking the inner product of (o) with the unit vector e_k thus gives

$t_k=\mathbf{t}\left({\mathbf{e}}_i\right)n_i·{\mathbf{e}}_k=T_{\mathit{ki}}{n}_i.$

  (p)

Equation (p) is the Cartesian component form of

$t=\mathbf{Tn}\text{or}\mathbf{t}\left(\mathbf{x},t,\mathbf{n}\right)=\mathbf{T}\left(\mathbf{x},t\right)\mathbf{n}.$

Recall from Section 4.4 that the heat flux h out of the surface $\partial \mathcal{P}$ is, in general, a function of position x, time t, and the geometry of the surface, i.e.,

$h=\tilde{h}\left(\mathbf{x},t,\text{geometry of surface}\right).$

We restrict our attention to a particular type of heat flow through the boundary $\partial \mathcal{P}$, namely, one such that the heat flux h is the same for all like-oriented surfaces with a common tangent plane at x and t, so (4.29) becomes

$h=\tilde{h}\left(\mathbf{x},t,\mathbf{n}\right).$

We further assume that h is a continuous function of x, t, and n. Then, as a consequence of the conservation laws of mass (4.11a), linear momentum (4.11b), and energy (4.19), and our boundedness assumptions, it can be shown (refer to Problem 4.4) that the dependence of h on n in (4.30) must be linear, i.e.,

$\tilde{h}\left(\mathbf{x},t,\mathbf{n}\right)=\tilde{\mathbf{q}}\left(\mathbf{x},t\right)·\mathbf{n}\text{or}h=\mathbf{q}·\mathbf{n}.$

The vector q in (4.31) is called the heat flux vector.